1. Field of the Disclosure
The present disclosure relates generally to determining geological properties of subsurface formations using Nuclear Magnetic Resonance (“NMR”) methods for logging wellbores, particularly for representing NMR echo trains by a limited number of functional parameters, enabling efficient transmission of echo train from a downhole location.
2. Description of the Related Art
NMR methods are among the most useful non-destructive techniques of material analysis. When hydrogen nuclei are placed in an applied static magnetic field, a small majority of spins are aligned with the applied field in the lower energy state, since the lower energy state in more stable than the higher energy state. The individual spins precess about the axis of the applied static magnetic field vector at a resonance frequency also termed as Larmor frequency. This frequency is characteristic to a particular nucleus and proportional to the applied static magnetic field. An alternating magnetic field at the resonance frequency in the Radio Frequency (RF) range, applied by a transmitting antenna to a subject or specimen in the static magnetic field transfers nuclear spins into a coherent superposition of the lower energy state and the higher energy state. In this superposition state the magnetization of the spins precesses about the axis of the static magnetic field vector and therefore induces an oscillating voltage in a receiver antenna even after the transmitted field is switched off, whose amplitude and rate of decay depend on the physicochemical properties of the material being examined. The applied RF field is designed to perturb the thermal equilibrium of the magnetized nuclear spins, and the time dependence of the emitted energy is determined by the manner in which this system of spins looses coherence and returns to equilibrium magnetization. The return is characterized by two parameters: T1, the longitudinal or spin-lattice relaxation time; and T2, the transverse or spin-spin relaxation time.
Measurements of NMR parameters of fluid filling the pore spaces of earth formations such as relaxation times of the hydrogen spins, diffusion coefficient and/or the hydrogen density is the basis for NMR well logging. NMR well logging instruments can be used for determining properties of earth formations including the fractional volume of pore space and the fractional volume of mobile fluid filling the pore spaces of the earth formations.
One basic problem encountered in NMR logging or MRI (imaging) is the vast amount of data that has to be analyzed. In well logging with wireline instruments, the downhole processing capabilities are limited as is the ability to transmit data to an uphole location for further analysis since all the data are typically sent up a wireline cable with limited bandwidth. In the so-called Measurement-while-drilling methods, the problem is exacerbated due to the harsh environment in which any downhole processor must operate and to the extremely limited telemetry capability: data are typically transmitted at a rate of no more than twenty bits per second.
A second problem encountered in NMR logging and MRI is that of analysis of the data. As will be discussed below, the problem of data compression and of data analysis are closely inter-related.
Methods of using NMR measurements for determining the fractional volume of pore space and the fractional volume of mobile fluid are described, for example, in Spin Echo Magnetic Resonance Logging: Porosity and Free Fluid Index Determination, M. N. Miller et al, Society of Petroleum Engineers paper no. 20561, Richardson, Tex., 1990. In porous media there is a significant difference in the T1 and T2 relaxation time spectra of the fluids mixture filling the pore space. Thus, for example, light hydrocarbons and gas may have T1 relaxation time of about several seconds, while T2 may be thousand times less. This phenomenon is due to diffusion effect in internal and external static magnetic field gradients. Internal magnetic field gradients are due to magnetic susceptibility difference between rock formation matrix and pore filling fluid.
Since oil is found in porous rock formations, the relationships between porous rocks and the fluids filling their pore spaces are extremely complicated and difficult to model. Nuclear magnetic resonance is sensitive to main petrophysical parameters, but has no capabilities to establish these complex relationships. Oil and water are generally found together in reservoir rocks. Since most reservoir rocks are hydrophilic, droplets of oil sit in the center of pores and are unaffected by the pore surface. The water-oil interface normally does not affect relaxation, therefore, the relaxation rate of oil is primarily proportional to its viscosity. However, such oil by itself is a very complex mixture of hydrocarbons that may be viewed as a broad spectrum of relaxation times. In a simple case of pure fluid in a single pore, there are two diffusion regimes that govern the relaxation rate. Rocks normally have a very broad distribution of pore sizes and fluid properties. Thus it is not surprising that magnetization decays of fluid in rock formations are non-exponential. The most commonly used method of analyzing relaxation data is to calculate a spectrum of relaxation times. The Carr-Purcell-Meiboom-Gill (CPMG) pulse sequence is used to determine the transverse magnetization decay. The non-exponential magnetization decays are fit to the multi-exponential form:
                              M          ⁡                      (            t            )                          =                              ∑                          i              =              1                        L                    ⁢                                    m              ⁡                              (                                  T                                      2                    ⁢                                                                                  ⁢                    i                                                  )                                      ⁢                          ⅇ                                                -                  t                                /                                  T                                      2                    ⁢                                                                                  ⁢                    i                                                                                                          (        1        )            where M(t) represents the spin echo amplitudes, equally spaced in time, and the T2i are predetermined time constants, equally spaced on a logarithm scale, typically between 0.3 ms and 3000 ms. The set of m are found using a regularized nonlinear least squares technique. The function m(T2i), conventionally called a T2 distribution, usually maps linearly to a volumetrically weighted distribution of pore sizes.
The calibration of this mapping is addressed in several publications. Prior art solutions seek a solution to the problem of mathematical modeling the received echo signals by the use of several techniques, including the use of non-linear regression analysis of the measurement signal and non-linear least square fit routines. Other prior art techniques include a variety of signal modeling techniques, such as polynomial rooting, singular value decomposition (SVD) and miscellaneous refinements thereof, to obtain a better approximation of the received signal. A problem with prior art signal compressions is that some information is lost.
Inversion methods discussed in prior art are generally computationally intensive and still end up with a large number of parameters that have to be transmitted uphole. In particular, no simple methods have been proposed to take advantage of prior knowledge about the structure of the investigated material and the signal-to-noise (SNR) ratio of the received echo signal. Also, no efficient solutions have been proposed to combine advanced mathematical models with simple signal processing algorithms to increase the accuracy and numerical stability of the parameter estimates. Finally, existing solutions require the use of significant computational power which makes the practical use of those methods inefficient, and frequently impossible to implement in real-time applications.
U.S. patent application Ser. No. 11/845,983 of Thern et al. discloses a method which includes conveying a nuclear magnetic resonance (NMR) sensing apparatus into a borehole, using the NMR sensing apparatus for obtaining a signal indicative of the property of the earth formation, using a predetermined matrix to estimate from the signal a parametric representation of the relaxation of nuclear spins in terms of at least one basis function, telemetering the parametric representation to a surface location and, at the surface location, using the telemetered parametric representation to estimate the property of the earth formation. The signal may be a spin echo signal and representation of relaxation of nuclear spins may include a transverse relaxation time (T2) distribution. The at least one basis function may be a Gaussian function, and parametric representation may include a mean, a standard deviation, and an amplitude of the Gaussian function. Defining the predetermined matrix may be done by performing a regression analysis on synthetic NMR signals and/or NMR signals measured on samples having known properties. The dependent variable in the regression analysis may be a spin echo signal. The regression analysis may be a partial least-squares, a principal component regression, an inverse least-squares, a ridge regression, a Neural Network, a neural net partial least-squares regression, and/or a locally weighted regression. The determined property may be bound volume irreducible, effective porosity, bound water, clay-bound water, total porosity, a permeability, and/or a pore size distribution.
A potential drawback of the method of Thern is the lack of adaptability: the number of Gaussian functions used to characterize the T2 distribution and the matrix are predefined and may not be equally suitable for all types of earth formations and all types of pulse sequences used in acquisition of the data. These drawbacks are addressed in the present disclosure.